Demystifying Latschev’s Theorem

Sush Majhi
Data Science Program, George Washington University

Today’s Agenda

  • The problem of shape reconstruction
  • Vietoris–Rips complexes
  • Theorems of Hausmann and Latschev
  • Demystifying Latschev’s theorem
    • Jung’s theorem on manifolds
    • Sketch of proof (if time permits)
  • Questions

The Problem of Shape Reconstruction

  • Shape: A Shape is modeled as a metric space \((M,d_M)\).

    • general compact set

    • metric graph

    • for today, \(M\) is a closed Riemannian manifold.

  • Sample: A finite metric space \((X,d_X)\) close to \(M\).

    • small Hausdorff proximity if \(M\) is a Euclidean submanifold and \(X\subset\mathbb R^d\)

    • for today, small Gromov–Hausdorff distance.

  • Goal: Infer the topology of \(M\) from \(X\).

    • Estimate only the Betti numbers: number of connected components, cycles, voids, etc, of \(M\).

    • for today, construct a topological space \(\widetilde{M}\) from \(X\) to retain the topology of \(M\), i.e., \(M\simeq\widetilde{M}\).

Vietoris–Rips Complex

  • a metric space \((X,d_X)\)

  • a scale \(\beta>0\)

  • \(\mathcal{R}_\beta(X)\) is an abstract simplicial complex such that

    • each subset \(A\subset X\) of size \(k\) with diameter at most \(\beta\) is a \((k-1)\)-simplex.

Hausmann’s Theorem

Convexity Radius

  • the largest (sup) radius so that geodesic balls are convex.

    • for the circle, it is a quarter of the circumference

    • denoted by \(\rho(M)\)

    • \(\rho(M)>0\) for a compact manifold

Theorem (Hausmann 1995): For any \(0<\beta<\rho(M)\), the Vietoris–Rips complex \(\mathcal{R}_\beta(M)\) is homotopy equivalent to \(M\).

  • vertex set is the entire manifold \(M\)
  • Question: what about the Vietoris–Rips complex of a dense subset \(X\subset M\)?
    • more generally, for a sample with a small \(d_{GH}(M,X)\).

Finite Reconstruction Problem

Gromov–Hausdorff Distance:

  • provides the noise model for our reconstruction problem
  • similarity measure between abstract metric spaces \((X,d_X)\) and \((Y,d_Y)\)
  • Definition: \(d_{GH(X,Y)}=\inf d_H^Z(f(X),g(Y))\)
    • inf over metric spaces \((Z,d_Z)\) and isometries \(f:X\to Z\), \(g:Y\to Z\)

Latschev’s Theorem (Latschev 2001):

For every closed Riemannian manifold \(M\), there exists a positive number \(\epsilon_0\) such that for any \(0<\beta\leq\epsilon_0\) there exists some \(\delta>0\) such that for every metric space \(X\) with Gromov–Hausdorff distance to \(M\) less than \(\delta\), the Vietoris–Rips complex \(\mathcal R_\beta(X)\) is homotopy equivalent to \(M\).

Quantitative Latschev’s Theorem

Metric Graph Reconstruction (Majhi 2023b)

Let \((G,d_G)\) be a compact, path-connected metric graph, \((X,d_X)\) a metric space, and \(\beta>0\) a number such that \[3d_{GH}(G,X)<\beta<3\rho(G)/4.\] Then, \(\mathcal R_\beta(X)\simeq G\).

Riemannian Manifold Reconstruction (Majhi 2023a)

Let \((M,d_M)\) be a closed, connected Riemannian manifold. Let \((X,d_X)\) be a compact metric space and \(\beta>0\) a number such that \[ \frac{1}{\zeta}d_{GH}(M,X)<\beta<\frac{1}{1+2\zeta}\min\left\{\rho(M),\frac{\pi}{4\sqrt{\kappa}}\right\} \] for some \(0<\zeta\leq1/14\). Then, \(\mathcal R_\beta(X)\simeq M\).

  • \(\kappa\) is an upper bound on the sectional curvatures of \(M\)
  • uses barycentric subdivision and Jung’s theorem on manifolds

Jung’s Theorem

For a compact set \(A\subset\mathbb R^n\), there exists a unique circumcenter, and the circumradius \(\mathfrak{R}(A)\) is at most \(\sqrt{\frac{n}{2(n+1)}}\) times the diameter of \(A\).

Manifold: (Dekster 1997) Let \(M\) be a compact, connected, \(n\)–dimensional manifold. For any compact subset \(A\) with \(diam(A)<\min\{\rho(M),\pi/4\sqrt{\kappa}\}\), its circumcenter exists in \(M\). Moreover, its diameter \[\begin{equation}\label{eqn:Jung} diam(A)\geq\begin{cases} \frac{2}{\sqrt{-\kappa}}\sinh^{-1}\left(\sqrt{\frac{n+1}{2n}} \sinh\left(\sqrt{-\kappa}\;\mathfrak{R}(A)\right)\right),&\text{ for }\kappa<0\\ 2\mathfrak{R}(A)\sqrt{\frac{n+1}{2n}},&\text{ for }\kappa=0 \\ \frac{2}{\sqrt{\kappa}}\sin^{-1}\left(\sqrt{\frac{n+1}{2n}} \sin\left(\sqrt{\kappa}\;\mathfrak{R}(A)\right)\right), &\text{ for }\kappa>0\text{ and}\\&\mathfrak{R}(A)\in \left[0,\frac{\pi}{2\sqrt{\kappa}}\right] \end{cases} \end{equation}\]

Discussions

  • Euclidean submanifold (Majhi 2023a)
  • Geometric reconstruction using Shadow Complexes
  • Reconstruction beyond manifolds
    • \(CAT(\kappa)\) spaces
    • simplicial complexes embedded in \(\mathbb R^d\)

References

Dekster, B. V. 1997. “The Jung Theorem in Metric Spaces of Curvature Bounded Above.” Proceedings of the American Mathematical Society 125 (8): 2425–33. https://doi.org/10.1090/s0002-9939-97-03842-2.
Hausmann, Jean-Claude. 1995. “On the Vietoris-Rips Complexes and a Cohomology Theory for Metric Spaces.” In Prospects in Topology (AM-138), 175–88. Princeton University Press.
Latschev, J. 2001. “Vietoris-Rips Complexes of Metric Spaces Near a Closed Riemannian Manifold.” Archiv Der Mathematik 77 (6): 522–28. https://doi.org/10.1007/PL00000526.
Majhi, Sushovan. 2023a. “Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data.”
arXiv:2204.14234 [Math.AT]. https://doi.org/10.48550/ARXIV.2305.17288.
———. 2023b. “VietorisRips Complexes of Metric Spaces Near a Metric Graph.” Journal of Applied and Computational Topology, May. https://doi.org/10.1007/s41468-023-00122-z.